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Mathematics LibreTexts

7.3: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 7.3.1

Use the horizontal line test to determine if the function is one-to-one.

  1. clipboard_e00dcca2039e18db8fefb626195322c1a.png
  2. clipboard_e41a01f3ceb2fa0e71d04b9a4c5220ae3.png
  3. f(x)=x2+2x+5
  4. f(x)=x214x+29
  5. f(x)=x35x2
  6. f(x)=x2x23
  7. f(x)=x+2
  8. f(x)=|x+2|
Answer
  1. no (that is: the function is not one-to-one)
  2. yes
  3. no
  4. no
  5. no
  6. no
  7. yes
  8. no

Exercise 7.3.2

Find the inverse of the function f and check your solution.

  1. f(x)=4x+9
  2. f(x)=8x3
  3. f(x)=x+8
  4. f(x)=3x+7
  5. f(x)=6x2
  6. f(x)=x3
  7. f(x)=(2x+5)3
  8. f(x)=2x3+5
  9. f(x)=1x
  10. f(x)=1x1
  11. f(x)=1x2
  12. f(x)=54x
  13. f(x)=xx+2
  14. f(x)=3xx6
  15. f(x)=x+2x+3
  16. f(x)=7xx5
  17. f given by the table: x24681012f(x)371852
Answer
  1. f1(x)=x94
  2. f1(x)=x+38
  3. f1(x)=x28
  4. f1(x)=x273
  5. f1(x)=(x6)22=x27236
  6. f1(x)=3x
  7. f1(x)=3x52
  8. f1(x)=3x52
  9. f1(x)=1x+1=1+xx
  10. f1(x)=(1x)2+2=1+2x2x2
  11. f1(x)=5y+4=5+4yy
  12. f1(x)=5y+4=5+4yy
  13. f1(x)=2x1x
  14. f1(x)=6xx3
  15. f1(x)=23xx1
  16. f1(x)=5x+7x+1
  17. x371852f1(x)24681012

Exercise 7.3.3

Restrict the domain of the function f in such a way that f becomes a one-to-one function. Find the inverse of f with the restricted domain.

  1. f(x)=x2
  2. f(x)=(x+5)2+1
  3. f(x)=|x|
  4. f(x)=|x4|2
  5. f(x)=1x2
  6. f(x)=3(x+7)2
  7. f(x)=x4
  8. f(x)=(x3)410
Answer
  1. restricting to the domain D=[0,) gives the inverse f1(x)=x
  2. restricting to the domain D=[5,) gives the inverse f1(x)=x15
  3. restricting to the domain D=[0,) gives the inverse f1(x)=x
  4. restricting to the domain D=[4,) gives the inverse f1(x)=x+6
  5. restricting to the domain D=[0,) gives the inverse f1(x)=1x
  6. restricting to the domain D=[7,) gives the inverse f1(x)=3x7
  7. restricting to the domain D=[0,) gives the inverse f1(x)=4x
  8. restricting to the domain D=[3,) gives the inverse f1(x)=3+410x

Exercise 7.3.4

Determine whether the following functions f and g are inverse to each other.

  1. \(f(x)=x+3, \quad g(x)=x-3\),
  2. f(x)=x4,g(x)=4x,
  3. f(x)=2x+3,g(x)=x32,
  4. f(x)=6x1,g(x)=x+16,
  5. f(x)=x35,g(x)=5+3x,
  6. f(x)=1x2,g(x)=1x+2.
Answer
  1. yes (that is: the functions f and g are inverses of each other)
  2. no
  3. no
  4. yes
  5. no
  6. yes

Exercise 7.3.5

Draw the graph of the inverse of the function given below.

  1. clipboard_e69f5fbec9b18afa60ce97d8ba4527d2f.png
  2. clipboard_e85679e1eed9dfd1931c682ec9ba575e4.png
  3. clipboard_e7d09259a7bff75260981a24ab80377f7.png
  4. f(x)=x
  5. f(x)=x34
  6. f(x)=2x4
  7. f(x)=2x
  8. f(x)=1x2 for x>2
  9. f(x)=1x2 for x<2
Answer
  1. clipboard_e5bd00924d99bc517e3bba566ce83928e.png
  2. clipboard_e0b24b66918fef93e58f7ed6943572efd.png
  3. clipboard_e9fd9e5fafae2fc1117afd331cb759f09.png
  4. clipboard_e5c7558ce7229316f34c32afad0639acf.png
  5. clipboard_ee561d70a10bbd68c45d025d7df21c4c2.png
  6. clipboard_ef1e04538ab49cb5b5e99f8ee7d75fc3f.png
  7. clipboard_ed333d3b0157ca15d4191e03782d7118d.png
  8. clipboard_e8144ade3afe05c7c97a70b1813458b86.png
  9. clipboard_e59e8e7f62c8759d878096556fde3f2e3.png

This page titled 7.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.

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